Please help with this question (cumulative distribution function of X)

In summary, a cumulative distribution function, or CDF, is a mathematical function that shows the probability of a random variable being less than or equal to a specific value. It is calculated by taking the sum of all probabilities of the variable being less than or equal to a given value. The CDF differs from a probability distribution function (PDF) in that it provides the probability of the variable taking on a specific value, while the PDF gives the probability of the variable being within a certain range. The CDF can be used to find the probability of a specific event occurring by subtracting the CDF at the lower bound from the CDF at the upper bound. It can also be used to find the median of a set of data, which
  • #1
tiffyuyu
2
0
The probability density function of the lifetime of a certain type of electronic device
(measured in hours), X, is given by

f(x) = 10/x^2,
0,
x > 10;
elsewhere.

(a) Find the cumulative distribution function of X, namely F(x) and hence find
P(X > 20).
(b) What is the probability that of 6 such devices, at least 3 will function for at least 20 hours? What assumption did you make?
 
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There are characters in your post that need to be edited, and can you show what you have done so far so our helpers see where you are stuck?
 

FAQ: Please help with this question (cumulative distribution function of X)

What is a cumulative distribution function (CDF)?

A cumulative distribution function, or CDF, is a mathematical function that shows the probability of a random variable being less than or equal to a specific value. It is often used to describe the overall distribution of a set of data.

How is the CDF of a random variable X calculated?

The CDF of a random variable X is calculated by taking the sum of all probabilities of X being less than or equal to a given value. This can be represented by the equation F(x) = P(X ≤ x), where F(x) is the CDF of X and P(X ≤ x) is the probability of X being less than or equal to x.

What is the difference between a CDF and a probability distribution function (PDF)?

The CDF and PDF are both used to describe the distribution of a set of data, but they differ in the information they provide. The CDF gives the probability of a random variable being less than or equal to a specific value, while the PDF gives the probability of the random variable taking on a specific value.

How can a CDF be used to find the probability of a specific event occurring?

The CDF can be used to find the probability of a specific event occurring by subtracting the CDF at the lower bound from the CDF at the upper bound. This can be represented by the equation P(a < X ≤ b) = F(b) - F(a), where P(a < X ≤ b) is the probability of X being between a and b, and F(a) and F(b) are the CDFs at a and b, respectively.

Can the CDF be used to find the median of a set of data?

Yes, the median of a set of data can be found using the CDF. The median is the value at which the CDF is equal to 0.5. This means that 50% of the data falls below the median and 50% falls above it.

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